[Physics.Soc-Ph] 4 Jun 2020 5 Last Words 7

Recent advances in opinion propagation dynamics: A 2020 Survey

Hossein Noorazar∗1 1Washington State University, Pullman, Washington, United States of America

Abstract Opinion dynamics have attracted the interest of researchers from different fields. Local interactions among individuals create interesting dynamics for the system as a whole. Such dynamics are important from a variety of perspectives. Group decision making, successful marketing, and constructing networks (in which consensus can be reached or prevented) are a few examples of existing or potential applications. The invention of the Internet has made the opinion fusion faster, unilateral, and on a whole different scale. Spread of fake news, propaganda, and election interferences have made it clear there is an essential need to know more about these dynamics. The emergence of new ideas in the field has accelerated over the last few years. In the first quarter of 2020, at least 50 research papers have emerged, either peer-reviewed and published or on pre-print outlets such as arXiv. In this paper, we summarize these ground-breaking ideas and their fascinating extensions and introduce newly surfaced concepts. Keywords:— Opinion dynamics, social dynamics, social interaction, consensus, polarization

Contents

1 Introduction 1

2 Preliminaries 2

3 Milestones 2 3.1 Continuous opinion space models ...... 2 3.1.1 DeGrootian models ...... 2 3.1.2 Bounded conﬁdence models ...... 3 3.2 Discrete opinion space models ...... 3 3.2.1 Galam model ...... 3 3.2.2 Sznajd model ...... 3 3.2.3 Voter model ...... 4

4 Milestones’ extensions 4 4.1 Stubborn agents ...... 4 4.2 Biased agents ...... 4 4.3 Opinion manipulation ...... 4 4.4 Power evolution ...... 5 4.5 Repulsive behavior ...... 5 4.6 Noisy models ...... 6 4.7 Interrelated topics ...... 6 4.8 Expressed vs. private opinions ...... 6 arXiv:2004.05286v2 [physics.soc-ph] 4 Jun 2020 5 Last words 7

6 New questions 7

7 Conclusions 8

8 Acknowledgments 8

1 Introduction ied the eﬀect of group pressure on social dynamics. French in 1956 was among the ﬁrst researchers to devote attention to opinion dynamics; A Formal Theory of Social Power [2]. Opinion dynamics studies propagation of opinions in a net- In 1964 Abelson [3] proposed a continuous-time model. A work through interactions of its agents. Modeling opinion decade later, in 1974, DeGroot established one of the sim- dynamics goes back a few decades. Asch in 1951 [1] stud- ∗[email protected]

1 plest models [4], which has become one of the most well An agent that does not change its opinion over time is re- known. Two years later Lehrer [5] also developed his model ferred to as a fully-stubborn agent. An agent that weighs its that is identical to that of DeGroot and an extensive discus- initial opinion, i.e., takes its initial opinion into account, in sion by Lehrer and Wagner are given in [6]. Another pioneer all interactions over time is referred to as a partially-stubborn and interesting work is [7] by Latané where he explored the agent. If an agent is not stubborn, it is called a non-stubborn interactions from a psychology angle and reviewed some of agent. In the bounded confidence models, we say an agent the earlier works such as group pressure, immitation, and is closed-minded if its confidence radius is smaller than that effects of newspapers. of others. Opinion dynamics take different shapes depending on the nature of the topic under consideration and the purpose of interactions. For example, the topic could be liking or dis- 3 Milestones liking a certain food such as fish; here, binary opinion dynamics comes into play [8–10]. Sometimes the opinion can In this section, we present the main models that have in- be represented by continuous variables. For example, the spired a tremendous amount of research. We start with the extent to which one supports a cause. Over the years, a models in which the opinion space is continuous and then few different models for continuous-opinions have been pro- present the models in which the opinion space is discrete. posed [2, 11, 12]. Either as an abstract idea or in real life, one can think of a continuous-opinion space in which one 3.1 Continuous opinion space models must take discrete actions [13, 14]. For instance, in an election where each agent’s support for candidates falls on the In this section, we present the DeGroot model and its ma- continuous spectrum, each agent must still cast a discrete jor extension known as the Friedkin-Johnsen model. Next, vote. we examine the bounded confidence models in which agents In short, opinion dynamics can be explained as follows. interact with those whose opinions are close enough to that Agents start with an initial opinion. Connected agents in- of their own. teract and update their opinion by a given clear update rule. This process is carried on until a termination criterion is met. 3.1.1 DeGrootian models An example of termination criteria is reaching a steady state in which agents do not change their opinion anymore. We begin with the simplest model, the DeGroot model. In this paper, we briefly introduce the main concepts and Moreover, since the Friedkin-Johnsen extension of the DeG- newly developed ideas of modeling opinion dynamics. Go- root model is well-known and has been studied extensively, ing into details is not the purpose of this survey. In Sec.3 we present it here as well. the major models are presented, then, in Sec.4 basic re- DeGroot model. The DeGroot model [4] is given by sults and certain extensions of the well-known models are reviewed. In Sec.5, we go through some models that have o(t) = Wo(t−1) = W2o(t−2) = ··· = Wto(0) (1) not been studied exhaustively but are interesting and have contributed novel concepts. Finally, the conclusions are pre- where W is a row-stochastic weight matrix and Wt is its t-th sented in Sec.7. power. The model is linear and traceable over time. Classi- cal linear algebra tools are sufficient to analyze this model. It has been very well studied and different extensions of it 2 Preliminaries exist. The DeGroot model is an iterative averaging model. If the network is connected then convergence is equivalent to The network of agents is denoted by G in which N agents consensus (Fig.1). Berger [22] showed the DeGroot model are present. Let A be the adjacency matrix where Aij = 1 will reach consensus if and only if there exists a power t of if agents i and j are connected and Aij = 0 otherwise. The the weight matrix for which Wt has a strictly positive col- row-stochastic influence matrix is denoted by W where Wij umn. Let W be such a matrix; then, the consensus opinion is the level of influence of agent j on agent i; 0 ≤ Wij ≤ 1. ∗ (0) T is given by o = hÀ, o i where `W is the left eigenvector Let us define the opinion space to be the set of all pos- of W associated with 1, constrained to h1N , Ài = 1. sible opinions denoted by O. Examples of opinion space are Friedkin-Johnsen model. One of the major exten- O = {0, 1}, {1, 2, . . . m}, [0, 1]. The opinion of agent i at sions of the DeGroot model was introduced by Friedkin and (t) time t is denoted by oi . The state of the system at time t Johnsen [23, 24] and is known as the Friedkin-Johnsen (FJ) (t) (t) (t) (t) o o ··· o is denoted by o = 1 2 N . model. Since it has functioned as a ground-breaking model, When the system reaches equilibrium, we say the system we include it here instead of in Sec.4. In the FJ model, the has converged. The convergence state may be consensus, idea of stubborn-agents is added to the DeGroot model: polarization, or fragmentation. Polarization is the state in which there are only two clusters of agents, and fragmenta- o(t+1) = DWo(t) + (I − D)o(0) (2) tion is the state there are more than two clusters. Before moving to the next section, we would like to men- where D = diag([d1, d2, . . . , dN ]) with entries that spec- tion that there is no convergence on terminology in the liter- ify the susceptibility of individual agents to influence, i.e., ature. For example, consider an agent that does not change (1 − di) is the level of stubbornness of agent i. For a fully- its opinion over time; some papers refer to such an agent as stubborn agent 1 − di=1, for a partially-stubborn agent a leader [15], others as a media [16], some as an stubborn 0 < 1 − di < 1 and for a non-stubborn agent 1 − di = 0. W agent [17] and a few as an inflexible agent [18,19]. A closed- is a row stochastic influence matrix. The convergence and minded agent is referred to someone who does not change its stability of the FJ model are studied in [25]. opinion in [20] and in some papers, a closed-minded agent is an agent whose confidence radius is small compared to These two models are the main two DeGrootian models. other agents [21]. We adhere to the following definitions. We now move on to the bounded confidence models.

2 Figure 1: Evolution of the DeGroot model from initial proﬁle to consensus.

3.1.2 Bounded confidence models Hegselmann-Krause model. The most well-known synchronous version of BCM is given by Hegselmann- In this section, we look at bounded confidence models. A Krause [26]. The (most common and simplest) update rule bounded confidence model (BCM) is a model in which agents of the HK model is given by: ignore the ideas that are too far from their own. The well- known pairwise BC model is given by Deffuant et al. [11] and (t+1) 1 X (t) oi = (t) oj (4) is called the DW model, while the most well-known syn- |N | (t) i j∈N chronous version is given by Hegselmann and Krause [26], i (t) and is called the HK model. where Ni is the set of neighbors of agent i at time t, i.e., the Deffuant-Weisbuch model. The celebrated DW (t) (t) set of agents for whom we have |oj −oi | ≤ r, including i it- model of Deffuant et al. [11] is defined by the following rule: self. Reference [26] includes the analyses of convergence and ( (t+1) (t) (t) (t) consensus for the HK model. Bhattacharyya et al. [34] study oi = oi + µ . (oj − oi ) (t+1) (t) (t) (t) (3) convergence properties of a multidimensional HK model. oj = oj + µ . (oi − oj ) where µ is the so-called learning rate that usually lies in 3.2 Discrete opinion space models (0, 0.5] to avoid crossover. The update takes place only if (t) (t) In this section, we focus on opinion models whose opinion |oj − oi | ≤ r where r is called confidence radius. In [11] all agents share the same confidence radius r and the same space is discrete. Variations of the Ising model provide ex- learning rate µ. Said differently, the system is homogeneous amples with binary opinion space. A discretized version of in both r and µ. Obvious variations can be achieved by DW [35] is another example. These models have applica- introducing a heterogeneous confidence radius to the sys- tions in real life. At least two studies used binary opinion tem, adding asymmetry in the confidence radius, or even models to explain Trump’s 2016 victory [36, 37]. an agent-specific time-varying confidence radius [27]. In- tuitively and in actuality, the confidence radius affects the 3.2.1 Galam model number of clusters at equilibrium. Consider the opinion In addition to Friedkin, who has left a large footprint in space O = [0, 1] and confidence radius r to be 1. Then, this field since 1986 [38], Galam has spent more than 35 each agent can interact with any other agent at all times, years studying opinion dynamics from a sociophysics per- and the subspace in which agents lie in will be contractive. (0) (0) spective [39,40] and his work has inspired other researchers. In another scenario, let r < m = mini,j {|oi − oj |}; then, He has studied a range of different dynamics [41] includ- there will be no interaction, and thus, there will be N clus- ing “democratic voting in bottom-up hierarchical systems, ters of size 1. In fact, Fortunato [28] claims r = 0.5 is the decision making, fragmentation versus coalitions, terrorism critical confidence radius above which the agents come to and opinion dynamics.” Reference [41] reviews Galam’s a consensus. Moreover, the number of clusters at equilib- work prior to 2008 and further details can be found in his rium is approximately 1/2r [29]. Lorenz [21] investigates a book [42]. Here, we introduce some of the newer works re- heterogeneous (in confidence radius) case in which there are lated to the binary opinion space used in the Galam model. two groups of agents. One group is closed-minded, i.e., has In the Galam model, there are two opinions in the opinion a smaller confidence radius, and the other group is open- space. The update rule is as follows; (1) agents are randomly minded. It is shown in [21] that heterogeneity of confidence distributed in groups of size r, (2) each group uses majority radius helps consensus to be reached; the final state will be rule to update their opinion, then (3) agents are shuffled and a consensus when the confidence radius of the open-minded the cycle begins again at step (1). group is well below the aforementioned r = 0.5 for the homo- Gärtner and Zehmakan [43] address consensus time and geneous case. Chen et al. [30] investigate convergence prop- sensitivity of outcome as functions of initial state in the erties of a heterogeneous (in confidence interval) DW model. Galam model. An asymmetric DW model is discussed in [31]. Shang [32] has proposed a modified DW model where confidence ra- 3.2.2 Sznajd model diuses are assigned to edges, as opposed to agents. Equiv- alently, agent i trusts agent j and k differently; the conver- Ising models have a long history in statistical physics. Here, gence properties of such a model are studied by Shang [32]. we overview one of the well-known models of this kind in the Another work that studies the convergence properties of a field of opinion dynamics, namely the Sznajd model [44]. In modified DW is [33], in which the learning rate is a function the Sznajd model, N agents are sitting on a 1-dimensional of opinion difference. lattice. Opinion space is given by O = {−1, +1}. At a given

3 time t, two neighbors i and i + 1 are selected randomly. If given in [60] also has the stubbornness ingredient where stub- (t) (t) oi ×oi+1 = 1 then agents i−1 and i+2 adopt the direction born agents are called leaders and their power of influence of agents i and i + 1, otherwise, the agent i − 1 adopts the is discussed. Cheon and Morimoto [61] consider a Galam opinion of agent i and agent i + 2 adopts the opinion of its model that includes balancer agents who oppose stubborn selected neighbor, agent i + 1. Steady states of such a model agents. Contrarian agents are specific to the Galam model, have all agents in agreement at either +1 or -1 or a stale- hence, we include them here. Galam and Cheon [62] inves- mate. The time needed to reach equilibrium is discussed tigate the effect of asymmetry in contrarian behavior which in [44] through Monte Carlo simulations. Some results from is an extension of [18]. the original Sznajd model and the Sznajd model on a com- Stubborn agents in voter model. References [63] plete graph are presented in [45]. and [64] explore the role of stubborn agents in a voter model Phase transition phenomena in the Sznajd model with and a noisy voter model, respectively. Yildiz et al. [65] exam- the presence of anticonformists in complete graphs are ex- ine the effect of stubborn agents with opposing views on the amined by [46, 47]. Calvelli et al. [47] also consider 2D and convergence of the system. Mukhopadhyay et al. [10] inves- 3D lattices. To learn more about the Sznajd model please tigate the effect of biased agents in both voter and majority- see [48, 49] rule dynamics. They also add stubbornness to the majority- rule case. The bias and stubbornness are implemented by 3.2.3 Voter model updating probability. They study the relationship between the size of the network and (1) consensus time, and (2) prob- In a voter model, opinion space is binary. At a given time ability of consensus. t, a random agent, i, is chosen. Then, i chooses a random neighbor and adopts the state of the neighbor. 4.2 Biased agents The voter model on regular lattices has been studied extensively. There are also variations of the voter model on Biased agents are more open to agents’ that hold similar different network topologies. Sood and Redner and [50] in- opinions to themselves as opposed to others, i.e., the ho- vestigate the voter model on a heterogeneous graph, and [51] mophily quality. explores the dynamics and convergence time of the voter Biased agents in DeGroot model. Dandekar et model on a graph with two cliques. The influence of an ex- al. [66] incorporate the idea of biased agents in the DeGroot ternal source is investigated in [52]. The impact of “active model and turn it into a nonlinear model. Xia et al. [67] links” on the convergence of the voter model is investigated provide some analysis for equilibria in such a model. in [53]. To learn more about voter models please see [54]. Biased agents in DW model. In the DW model, a Examples of the other extensions are given below. pair of agents are chosen randomly. Sîrbu et al. [68] modify the DW model to add the bias ingredient. In this model, agent i is chosen randomly and then the interaction partner 4 Milestones’ extensions j is chosen by a probability function that depends on the difference of opinion. The closer the opinion of agent j is to the We are ready to investigate some of the fascinating exten- opinion of i, the more the probability of interaction between sions of the reviewed models in the previous section. them. Convergence properties and network size effects are addressed by [68]. 4.1 Stubborn agents Biased agents in HK model. Chen et al. [69] take the modified HK model of Fu et al. [70] and extend it to Stubborn agents in DeGroot model. One major modi- include biased agents. They call the new model the “Social- fication of the DeGroot model known as the FJ model adds Similarity-Based HK model.” In this model, for two agents stubbornness and was presented previously. However, here to interact not only do they need to hold close opinions but we introduce other versions that are new and have not stud- the criteria of social similarity also must be met, i.e., their ied extensively. Abrahamsson et al. [17] study the effect of other attributes need to be close as well. Social similarity the presence of fully-stubborn agents in the DeGroot model. can be measured by considering different attributes such as Wai et al. [55] propose “an active sensing method to estimate age, education, and other traits. the relative weight (or trust) agents place on their neighbors”’ and explore the role of stubborn agents in such an 4.3 Opinion manipulation environment. Zhou et al. [56] study the effect of partially- stubborn agents on a modified DeGroot model. In their Opinion manipulation is fascinating for different reasons. altered model, an agent not only takes the opinions of its Maximizing the number of customers in a market or inter- neighbors into account but also takes the opinions of its fering with another country’s election are two examples of neighbors’ neighbors into account as well. opinion manipulation. Below, we review some of the pro- Stubborn agents in DW and HK. The effect of Stub- posed models. born agents in DW and HK is explored in [57] and [16], re- Opinion manipulation in DeGroot model. There spectively. In the latter, stubborn agents are labeled as me- are a few works about how to make a network reach a con- dia. They investigate “how the number of media accounts sensus and further still, how to influence the agents toward and the number of followers per media account affect the a predetermined opinion [71–73]. Dong et al. [71] propose media impact.” Moreover, one of their novel contributions a network modification, i.e., adding a minimum number of is “content quality.” edges to the network, to reach consensus in the DeGroot Stubborn agents in Galam model. References [18, model. Zhou et al. [56] consider the manipulation of public 58] study the effect of inflexible agents in the Galam model. opinion in a modified DeGroot model. Hegselmann et al. [74] Another work [59] is an extension of the Galam model in use one strategic agent who can change its opinion freely to which they study how the minority wins against the major- steer as many as possible agents towards a predetermined ity in scenarios such as the US election in 2016. The model interval.

4 Opinion manipulation in FJ model. To the best of study the evolution of power in a network. In the work of Jia our knowledge, there is no study to influence agents toward et al. [87], the power of influence of agents evolves over a se- consensus in the FJ model. Previously, a missing piece for quence of topics where the dynamic of each topic discussion manipulation of agents in models was preventing a network is governed by the DeGroot model. Suppose the network dis- from reaching a consensus. A very recent study, Gaitonde et cusses a sequence of topics s = 0, 1, 2, ··· one after another al. [75], investigates adversarial manipulation of a network to where the dynamic of each discussion is governed by the De- prevent it from consensus where the dynamics are governed Groot model. The weight matrix for each topic depends on by the FJ model. the outcome of the previous topic: Opinion manipulation in DW model. Pineda and (t+1) (t) Buendía [72] investigate the effect of mass media in both o (s) = W(s)o (s) (5) DW and HK models. They consider heterogeneous (in con- In Eq.5 the weight matrix W(s) depends on the outcome of fidence radius) cases for both DW and HK and study con- the topic s−1. This model is known as the DeGroot–Friedkin ditions under which the effect of mass media is maximized. model. The relation between social power and centrality Another example of affecting the network’s opinion is pre- ranking is established in [87]. Moreover, the conditions un- sented in [76]. der which a democratic or autocratic structure is formed are Opinion manipulation in HK model. Standard tools discussed. in linear algebra enable one to understand the dynamics of Kang et al. [88] use a two-layer network to explore the the DeGroot model. Such results help to manipulate the evolution of social power in the DeGroot-Friedkin model. opinion of the agents by modifying the topology of the net- Convergence properties of such a model are provided. The work [71]. However, this is not the only way to manipulate weight matrix W(s) is decomposed into a sum of two ma- the network’s final state. Brooks and Porter [16] use media trices W(s) = D(s) + (I − D(s))C where C is called a rel- to manipulate the outcome of the discussion in a network; n ative influence matrix. C is row-stochastic, irreducible with “We maximize media impact in a social network by tuning a zero diagonal. In the case of [88], there are two relative the number of media accounts that promote the content and influence matrices, one for each layer. It is shown in [87] the number of followers of the accounts.” We mentioned that for the DeGroot-Friedkin model, a democratic configu- Hegselmann et al. [74] extended the DeGroot model to in- ration will be reached if and only if the relative matrix C is clude one strategic agent to manipulate other agents. This doubly-stochastic. In the case of a two-layer network of [88], paper also applies the same idea to the HK model as well. the democratic configuration will be reached if both of the Another work [77] investigates the role of a stubborn agent influence matrices are doubly-stochastic. They both have in an extension of the HK model. We include it in this sec- similar results for emerging autocratic configuration under tion since it investigates the space of parameters to study star topology. conditions under which the stubborn agent (external signal) The evolution of individuals’ power is further studied can maximize its influence in attracting other agents. They in [89–95]. Ye and Anderson [91] extended the DeG- define intensity of the signal as the number of times the root–Friedkin model by adding new characteristics to agents; signal is sent at time t. Interestingly, they discover higher “humbleness” and “unreactiveness.” In this extension, power intensities may have less effect in attracting other agents. evolution of agents is “distorted” by these new characteris- This result is similar to that of [72]. The intensity in case tics. These researchers study the existence and uniqueness of [72] is defined as the probability of interaction between a of equilibria, and convergence if it exists. Askarzadeh et stubborn agent and another agent in the DW model. al. [94] study the power evolution of the DeGroot–Friedkin Opinion manipulation in voter model. Gupta et model using probability and Markov chain theory. al. [78] propose strategies for manipulating the agents’ opin- Power evolution in FJ model. Tian et al. [95] study ion in a voter model. The influence maximization on a com- power evolution in the FJ model, including the properties plex network is given in [79], and influence maximization of equilibria and the conditions under which democracy is by considering the agents’ power of influence, the Influence achieved. Furthermore, it is shown that autocracy cannot Power-based Opinion Framework, is proposed in [80]. be achieved in the presence of stubborn agents. More on opinion manipulation. We can mention We mentioned earlier that stubborn agents are also called Refs. [81–84] as other examples of opinion manipulation. media or leaders. The reason is that they can influence Goyal and Manjunath [84] build on [85] and investigate a other agents and have a major impact on the final state of scenario in which two competing forces try to gain control the system. Equivalently, stubbornness translates into social of the network and maximize the number of their followers. power. This fact is not only observed in the FJ model, but Each “controller” has a budget constraint and Nash con- also in the DeGroot model (e.g. [96]). trol strategies are determined for each controller. Brede [86] investigates a rewiring model in which influencers try to maximize their impact. 4.5 Repulsive behavior The models we observed so far only support two types of 4.4 Power evolution behavior–attraction or indifference. Humans are more com- plicated. If the topic is sensitive then repulsive behavior We mentioned two properties of the final state in the DeG- emerges and causes polarization or fragmentation. Let us root model. If the network of agents is connected, i.e., the look at models that support such behavior. network does not consist of disjoint subgraphs, then the fi- Repulsion in the DeGroot model. Chen [97] adds a nal state is consensus and the consensus value is a weighted repulsive behavior to the model of Dandekar et al. [66]. The average of the initial opinions. These two properties were model proposed in [97] uses a single parameter– entrench- the motivation for the introduction of power evolution in ment parmeter– to capture both bias and backfire effect. the DeGroot model. Their model also supports polarization that previously did Power evolution in DeGroot model. Jia et al. [87] not exist in the original DeGroot model.

5 Repulsion in DW model. Repulsive behavior in a presence of noise is addressed in [124]. A modification of the modified DW model is discussed in [98–100]. The model HK [125] models uncertainty of agents. In this model, some proposed in [12] is based on minimizing interaction energy agents may have an opinion that is actually an interval, not between agents. The interaction energy is defined via poten- a single number. tial functions. The update rule in [12] is given by: Noise in Galam model. Hamann [126] explores noise  in a modified Galam model in a group of mobile agents with (t+1) (t) µ 0 (t) (t) (t) (t) the presence of contrarians.  oi = oi − 2 ψ (|oi − oj |)(oi − oj ) (6) Noise in Sznajd model. Sabatelli and Richmond [127]  (t+1) (t) µ 0 (t) (t) (t) (t)  oj = oj + 2 ψ (|oi − oj |)(oi − oj ) add noise to a modified Sznajd model where the updates are done in a synchronous fashion. One of their results in that With the proper choice of a potential function, this model they “ predict that consensus can be increased by the addi- collapses to the DW model (see Fig. 2a), or the model of tion of an appropriate amount of random noise.” Jager and Amblard [100] (see Fig. 2b). Using the potential Noise in voter model. One of the earliest noisy voter function in Fig. 2b, the model will support three types of be- models is [128], which employs standard statistical physics havior: attraction, indifference, and repulsion. The poten- techniques and examines the critical behavior of the system tial function given in Fig. 2c produces a modified DW model and its phase transition. References [129, 130] examine the with repulsive behavior. In these three potential functions role of noise in the voter model on complex networks. The the confidence radius is r = 0.3. role of “zealots”, i.e., fully-stubborn agents, in a noisy voter More on repulsive behavior. References [101–110] model is inspected in [64] where agents form a fully con- consider signed graphs and the concept of balance the- nected graph. ory [111] in their work of modeling antagonistic or repulsive behavior. In a signed graph each edge is labeled with a positive or negative sign, defining friendship or antagonistic 4.7 Interrelated topics relationships. Aghbolagh et al. [110] has implemented three It is rarely the case that a given issue exists in an isolated types of behavior–attraction, indifference, and repulsion–in environment. The change of opinion about one topic could a modified HK model. They show their new model can lead cause a change of opinion about another topic. For instance, to consensus, bipartite consensus, and clustering of opinions. a change of opinion about health can lead to a change of opinion about exercise and diet. This area has not yet seen 4.6 Noisy models much exploration. Below, we present the limited models of Noise is injected into models for different purposes. For in- this nature. stance, Mäs [112] uses noise to implement the idea of the Interrelated topics in FJ model. In 2016, two inde- tendency for uniqueness in the model of Durkheim [113]. pendent works [12,131] proposed novel ideas for the dynam- The strength of noise in this model increases as the clusters ics of interrelated (coupled or interdependent) topics. The grow in size. Other forms of noise are implemented to model model in [12] is novel and does not fall under the umbrella different traits of humans’ behavior. Noise can be used to of classical models; however, the Ref. [131] is a generaliza- model the death or birth of an agent, and to mimic internal tion of the FJ model. Friedkin et al. [131] revisit the idea thoughts or interactions with external sources such as media of interdependent topics in the multidimensional FJ model or books. Below we review some of the noisy models. in [25]. Tian and Wang [132] introduce the idea of sequen- Uncertainty in DeGroot model. A modified DeG- tially dependent topics in which each topic is discussed in a root model, taking into account uncertainty of agents en- sequence and the outcome of topic s affects the dynamics of coded as intervals is given in [98]. the discussion of topic s + 1 where each topic’s dynamic is Noise in DW model. One can argue humans do not governed by the FJ model. have a sharp threshold like a confidence radius for accept- Interrelated topics in DW model. Fei et al. [133] ing or rejecting other ideas. Grauwin and Jensen [114] use propose a model for interdependent topics where interactions random noise in the DW model to kill the aforementioned are pairwise and follow the bounded confidence concept. sharp threshold. in [114], two agents interact with a prob- More on interrelated topics. Ahn et al. [134] propose ability that depends on the difference of opinion of the two a novel opinion model with interrelated topics. Let ` and s agents–an interaction noise. Another type of noise intro- to be two given coupled topics. in [12], for example, the induced in [114] is reminiscent of the death of a person and teraction between agents is based on a given topic. Suppose birth of another; an agent changes its opinion at time t to agents i and j interact about topic ` and update their opin- a random opinion with some probability. Pineda et al. [115] ion. Consequently, by internal thoughts due to the coupling investigate another type of noise in the DW model: “Indi- of ` and s, their opinion about s will be updated as well, viduals are given the opportunity to change their opinion, despite the fact that they did not discuss topic s. However, with a given probability, to a randomly selected opinion in- in [134] it is possible that topic ` of agent i is coupled with side an interval centered around the present opinion.” Ref- topic s of agent j. Other novel models have recently been erences [116–120] also investigate the noise effect in models proposed, such as [135], Bayesian learning model [136], a inspired by the DW model. model based on Achlioptas Process [137] and a model based Noise in HK model. Su et al. [121] introduce noise on Latané’s social impact theory [138], to name a few. to the homogeneous HK model and show how it can help the formation of consensus. A recent study investigates the 4.8 Expressed vs. private opinions role of environment and communication noise in the heterogeneous HK model [122]. Phase transition and conver- The expressed opinion of agents is not always identical to gence time are studied in the presence of environmental noise their true internal belief. Social pressure can cause people in [122]. Another example of noise in the HK model is dis- to express an opinion that aligns with that of others while cussed in [123]. Nonlinear stability for the HK model in the contradicting their internally held belief. This concept was

6 (a) DW potential (b) Jager-Amblard potential (c) A repulsive DW potential

Figure 2: Potential function examples. By choosing the potential function in (a) the model of [12] produces the DW model, and the potential function given in (b) results in Jager-Amblard model. The potential function in (c) generates another simple modified DW model with repulsive behavior. proposed in 1990 by Nowak and Latané [139] which is based the synchronous HK model. In this model, there is a poten- on earlier work of Latané [7]. In this section, we present more tial to interact with agents beyond one’s confidence radius. recent models that consider the co-evolution of expressed More details about the model are given in [152]. Another and personal opinions. interesting work [20] studies convergence of a modified HK Expressed vs. private opinions in FJ. The dynamics model in which agents have inertia. References [70,153,154] of the co-evolution of expressed and private opinions (EPO) contain other examples of the study of the convergence prop- in the FJ model along with its convergence properties are erties of the HK model and its variations. Gang et al. [155] presented in [140]. investigate the final state of the heterogeneous (in confidence Expressed vs. private opinions in voter model. radius) HK model. References [141–143] explore different ideas about expressed Rubio et al. [156] have recently proposed a model for and private opinions in the voter model. anomaly detection in the Industrial Internet of Things ar- More on EPO. Some novel models explore the dynam- chitectures. Other examples of applications of opinion dy- ics of private and expressed opinions [144–147]; while they namics in engineering are given in [157] and [158], where the do not fall under the umbrella of well-known models, they later reference studies voting processes inspired by BCM. are worth mentioning. We mentioned agents may reveal an opinion that is differ- Physics has inspired different models of opinion dynam- ent from their internal true belief. in [140] the co-evolution ics, of course. There are several works based on the kinetic of the two (internal and revealed) opinions are studied. We theory of gases [159–169] where interactions are defined by have also talked about manipulating the agents to influence Boltzmann type equations. Applications of such models them toward a predetermined target. The model presented in other fields, such as economics, are found in the book by Afshar and Asadpour [148] is somewhere in between. by Pareschi and Toscani [170]. Düring et al. [171] inves- Their model is inspired by the DW model and includes some tigate the presence of leaders and Wang et al. [172] took informed agents who pretend their opinion is close to that of the effect of noise into account in these dynamics. Further- other agents. These informed agents influence other agents more, the mean-field theory has been employed to explore toward a predetermined target opinion. the landscape of dynamics [173–176]. The Ising model is Table.1 provides an overview of the material presented another tool that can be used when the opinion space is bi- in this paper. nary [37, 177–179], with applications in areas such as group decision making [44, 180].

Lastly, it is worthwhile mentioning the evolution of 5 Last words agents’ susceptibility to persuasion, which is examined in [181, 182]. Edge weights are used to implement the fre- Before closing the discussion, we would like to cover other quency of interaction between agents in [183,184]. For more interesting models that have not yet been studied exten- details about continuous-opinion-space models, we refer the sively and acknowledge the great eﬀorts of other researchers. reader to the tutorials in Refs. [185, 186].

Li et al. [149] propose an interesting model. Unlike BCMs, agents in their model of [149] interact if the dif- 6 New questions ference of opinion is larger than a threshold due to social pressure. In the model given in [150] there is a potential to While a great deal of progress has been made in the ﬁeld, interact with agents whose opinions outside of the conﬁdence there is still great potential for improvement. Humans do interval. not interact with all their neighbors simultaneously, unlike Zhang and Hong [151] propose two synchronous versions the DeGroot model. Even for a network of computers that of BCM in which not all neighbors of agent i participate can interact quickly and can follow a clear set of rules, there in the update of the opinion of agent i. Instead, several are physical limitations. Balanced graphs are used to model neighbors are selected randomly. These researchers are in- repulsive behavior based on principles such as “friend of my terested in the convergence properties of this model, which friend, is my friend” or “enemy of my enemy is my friend,” sits between the pairwise interaction in the DW model and which are not always true.

7 While some of the opinion dynamics models are designed lution, interrelated topics, noise, and expressed and private to model a certain trait (e.g., homophily) or are tailored opinions. Finally, we posed new questions for future explo- to create interesting dynamics (e.g., preventing consensus rations. by introduction of noise), these models are not universal. Hence, it would be interesting to shrink the gap between simplicity of theoretical models and complexity of humans’ 8 Acknowledgments behavior. Opinion dynamics could be used to detect fake-news re- We would like to acknowledge the insightful inputs of Mo- sources on social media. Detecting susceptible individuals hammad Hossein Namaki that immeasurably helped in the who might be attracted to terrorist groups via the Internet development of this manuscript. is another potential domain of work. It would be helpful to see more applications of opinion References dynamics in real world problems. More specifically, it would be fascinating to take advantage of opinion dynamics models [1] S E Asch. Effects of group pressure upon the modifica- to detect and flag computers or processors sending erroneous tion and distortion of judgments. In Groups, leadership or corrupted messages in computer buses. and men; research in human relations., pages 177–190. Model Objection References Carnegie Press, Oxford, England, 1951. DeGroot Convergence [22] [2] Jr. French, J. R. P. A formal theory of social power. Stubbornness [17, 23, 55, 56] Psychological Review, 63(3):181–194, 1956. Bias [66, 67] [3] Robert P Abelson. Mathematical models of the distri- Opinion manipulation [56, 71–73] bution of attitudes under controversy. Contributions Repulsion [97] to mathematical psychology, 1964. Power evolution [87–95] FJ Convergence [25] [4] Morris DeGroot. Reaching a consensus. Journal of Opinion manipulation [75] the American Statistical Association, 69(345):118–121, Power evolution [95] 1974. Interrelated topics [25, 131, 132] [5] Keith Lehrer. When rational disagreement is impossi- EPO [140] ble. Noûs, pages 327–332, 1976. DW Convergence [21, 28, 30–33] [6] Keith Lehrer and Carl Wagner. Rational consensus in Stubbornness [57] science and society: A philosophical and mathematical Bias [68] study, volume 24. Springer Science & Business Media, Opinion manipulation [72] 2012. Repulsion [98, 99] Interrelated topics [133] [7] Bibb Latané. The psychology of social impact. Amer- Noise [114–120] ican psychologist, 36(4):343, 1981. HK Convergence [26, 34] [8] Soham Biswas and Parongama Sen. Model of bi- Stubbornness [16, 77] nary opinion dynamics: Coarsening and effect of dis- Bias [69] order. Physical Review E-Statistical, Nonlinear, and Opinion manipulation [16, 74, 77] Soft Matter Physics, 80(2):4–7, 2009. Noise [121–124] [9] Fei Ding, Yun Liu, Bo Shen, and Xia-Meng Si. An Galam Convergence [43] evolutionary game theory model of binary opinion for- Stubbornness [18, 58–61] mation. Physica A: Statistical Mechanics and its Ap- Contrary [18, 62] plications, 389(8):1745–1752, 2010. Noise [126] [10] Arpan Mukhopadhyay, Ravi R. Mazumdar, and Rahul Voter Stubbornness [10, 63–65] Roy. Opinion dynamics under voter and major- Opinion manipulation [78–80] ity rule models with biased and stubborn agents. EPO [141–143] arXiv:2003.02885, 2020. Noise [64, 128–130] [11] Guillaume Deffuant, David Neau, Frederic Amblard, Table 1: Overview of materials presented in the paper. and Gérard Weisbuch. Mixing beliefs among interacting agents. Advances in Complex Systems, 03(01n04):87–98, 2000. 7 Conclusions [12] Hossein Noorazar, Matthew J. Sottile, and Kevin R. Vixie. An energy-based interaction model for popu- In this paper, we reviewed well-known models of opinion lation opinion dynamics with topic couplin. Interna- dynamics for both continuous and discrete opinion spaces. tional Journal of Modern Physics C, 29(11):1850115, In the continuous-opinion-space case, we reviewed the DeG- 2018. root model, and one of its major extensions, namely the FJ model. Afterwards, we presented the two major bounded [13] André C. R. Martins. Continuous opinions and dis- confidence models, namely the DW model and the HK crete actions in opinion dynamics problems. Interna- model. In the discrete-opinion-space case, we reviewed the tional Journal of Modern Physics C, 19(4):617–624, Galam model, the Sznajd model and the voter model. Sub- 2007. sequently, for the selected additional models, reviewed some [14] André C. R. Martins. Discrete opinion dynamics with extensions that added extra ingredients(s) to the original M choices. The European Physical Journal B, 93(1):1, model– stubbornness, bias, repulsive behavior, power evo- 2020.

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